Integral Calculus, Trigonometry

Trigonometry

The word trigonometry literrlay means the measurement of triangles. Given a right triangel, place its base along the x axis, and let its hypotenuse, the longer slanted side, meet the base at the origin. The base extends to the right of the origin, and the hypotenuse slants up and to the right. Let the hypotenuse and base form an angle of θ. Hence 0 < θ < π/2. Rescale the x and y coordinates by a common factor h. This does not change the ratio of the lengths of any two line segments in the plane; both lengths are divided by h. In particular, it does not change the ratio of any two sides of our triangle. Now the hypotenuse haslength 1. In other words, the hypotenuse is a radius that extends from the origin out to the unit circle. By the definition of angle, its coordinates are cos(θ), sin(θ). Therefore the cosine of an angle is always the length of the base divided by the length of the hypotenuse. Similarly, the sine is given by the opposite side divided by the hypotenuse.

Let's consider an angle of 45°, or π/4. This is half the quarter circle. By symmetry, the sine and cosine are the same. Since sine squared plus cosine squared = 1, sin(45°) = cos(45°) = sqrt(1/2) = 0.707.

In another example, consider 30°, or π/6. Reflect this through the x axis and put the two triangles together to form an equilateral triangle. Thus the opposite side is half the hypotenuse. This means sin(30°) = 1/2, and cos(30°) = sqrt(3)/2.

The tangent of θ is the sine divided by the cosine. Returning to our triangle, this is the opposite side divided by the base. Sometimes the base is called the adjacent side, so you will see tangent = opposite/adjacent. If you have trouble remembering what over what is what, try the following.

Sine    Cosine  Tangent 
Oliver  And     Olivia  
Have    Had     Algebra

Restrict to the first letters, then expand again to A=adjacent O=opposite H=hypotenuse. Personally I think it's easier just to remember the ratios, or picture the triangle embedded in the unit circle, but hey, if you like mnemonics, there it is.